Learning functional components of PDEs from data using neural networks
This work addresses the challenge of modeling PDEs with unmeasurable components for researchers in computational science, though it is incremental as it extends existing parameter-fitting workflows to functions.
The authors tackled the problem of recovering unknown functions in partial differential equations (PDEs) from data, showing that neural networks embedded in PDEs can approximate these functions with arbitrary accuracy, as demonstrated by recovering interaction kernels and external potentials from steady-state data in nonlocal aggregation-diffusion equations.
Partial differential equations often contain unknown functions that are difficult or impossible to measure directly, hampering our ability to derive predictions from the model. Workflows for recovering scalar PDE parameters from data are well studied: here we show how similar workflows can be used to recover functions from data. Specifically, we embed neural networks into the PDE and show how, as they are trained on data, they can approximate unknown functions with arbitrary accuracy. Using nonlocal aggregation-diffusion equations as a case study, we recover interaction kernels and external potentials from steady state data. Specifically, we investigate how a wide range of factors, such as the number of available solutions, their properties, sampling density, and measurement noise, affect our ability to successfully recover functions. Our approach is advantageous because it can utilise standard parameter-fitting workflows, and in that the trained PDE can be treated as a normal PDE for purposes such as generating system predictions.